Continuous-valued analog circuits for Lukasiewicz implication (.fwdarw.) and negated implication (.about..fwdarw.) are the basis for analog array processors called Lukasiewicz logic arrays (.English Pound.LAs).
Lukasiewicz logic (hereafter .English Pound.) has a denumerably infinite number of truth values. (.English Pound..sub.1 is replaced by .English Pound. to avoid confusion when using .English Pound..sub.n to denote subsets of .English Pound..sub.1). .English Pound. describes the class of ideal analog circuits that have an infinite maximum precision (see FIG. 1). Real analog circuits, which have a finite maximum precision, are classified by the number of data values encoded on individual wires in a circuit. A circuit is described by a subset of .English Pound. designated as .English Pound..sub.n, where n specifies a whole number of truth values. .English Pound..sub.2 is the most familiar subset of .English Pound.: it includes the Boolean logic that describe binary digital circuits. Discrete multiple valued circuits are described by subsets of .English Pound. with 2&lt;n.ltoreq.16. Continuous-valued analog circuits are described by subsets of .English Pound. with 2.ltoreq.n.ltoreq.2.sup.p, where p is the maximum precision of the circuit in bits. Typically, p falls in the range 6.ltoreq.p.ltoreq.12.
Continuous-valued analog .English Pound.LAs have a dual logical and algebraic semantics that makes them capable of both symbolic and numeric computation. Fuzzy controllers, neural networks, inference engines and general-purpose analog computers (GPACs) can be implemented with .English Pound.LAs.
An .English Pound.LA is organized as an H-tree array that implements a sentence schema of .English Pound.. The processing elements of the .English Pound.LA correspond to implications or negated implications in the sentence schema. Three implication cells are shown in FIG. 2. Lukasiewicz implication is defined by the valuation function v(.alpha..fwdarw..beta.)=min(1,1-.alpha.+.beta.) and represented by the symbol ".fwdarw." or .OR left.. Negated implication has a valuation function defined as v(.alpha..about..fwdarw..beta.)=max(0, .alpha.-.beta.) and represented by the symbol ".about..fwdarw." or . Negated implication is identical to bounded difference (.THETA.), but indicates its relation to implication: .about..fwdarw..ident..about.(.alpha..fwdarw..beta.). The term implication is occasionally used to refer to both functions.
Applications for Lukasiewicz logic arrays include fuzzy controllers, neural networks, tautology checkers for .English Pound., and general-purpose analog computers (GPACs). Arithmetic and logical functions can be defined using implication or negated implication. A specific processor can be constructed by mapping arithmetic and logical functions with varied precision to one or more .English Pound.LAs.
Analog computers of the 1950's and 1960's used diodes to build special function generators. However, the link between these special function generators an Lukasiewicz logic was not recognized, although Wilkinson described an "analog diode logic" that corresponds with Lukasiewicz logic in an IEEE Transactions on Electronic Computers article in April of 1963. Corver Mead's innovations in analog VLSI for neural systems, the use of Lukasiewicz logic as a foundation for approximate reasoning systems, and the advent of practical .English Pound.LAs point to the renewed growth of analog computing after a 30-year hiatus.
Architectures for these applications cannot be implemented with known prototype .English Pound.LAs, which contain too few implication circuits. The Achernar .English Pound.LA compiler (available from the Computer Science Department of Indiana University in Bloomington, Ind.) typically outputs sentences containing hundreds of implications, with many internal and few external connections. .English Pound.L9, a prototype .English Pound.LA, has only 32 implications per chip. Practical architectures built with .English Pound.LAs require at least 1,000 implication circuits on a chip. .English Pound.LAs of this density can be built with the area-efficient implication and negated implication circuits described in this application.